MinT - Best Known (63−15, 63, s)-Nets in Base 9

Best Known (63−15, 63, s)-Nets in Base 9

(63−15, 63, 1876)-Net over F₉ — Constructive and digital

Digital (48, 63, 1876)-net over F₉, using

9³ times duplication [i] based on digital (45, 60, 1876)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁶⁰, 1876, F₉, 15, 15) (dual of [(1876, 15), 28080, 16]-NRT-code), using
  - OOA 7-folding and stacking with additional row [i] based on linear OA(9⁶⁰, 13133, F₉, 15) (dual of [13133, 13073, 16]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁶⁰, 13134, F₉, 15) (dual of [13134, 13074, 16]-code), using
      - trace code [i] based on linear OA(81³⁰, 6567, F₈₁, 15) (dual of [6567, 6537, 16]-code), using
        constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
        linear OA(81²⁹, 6562, F₈₁, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(81²⁵, 6562, F₈₁, 13) (dual of [6562, 6537, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(81¹, 5, F₈₁, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(63−15, 63, 14882)-Net over F₉ — Digital

Digital (48, 63, 14882)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−15, 63, large)-Net in Base 9 — Upper bound on s

There is no (48, 63, large)-net in base 9, because

13 times m-reduction [i] would yield (48, 50, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]